Minimal Coupling

Prerequisites: Gauge Symmetry. Full path (14 pages)

Intuitive
Concrete
Abstract
Why is it interesting?

Intuitive

Minimal coupling is the standard recipe for letting charged matter feel a gauge field such as the electromagnetic field. We start from the equations of a free particle or a free field and replace every ordinary derivative by a covariant derivative, which contains the gauge field. Nothing else is added.

The reason this works is gauge symmetry. The phase of a charged quantum field can be changed independently at every point without changing the physics. An ordinary derivative compares the field at neighbouring points and is spoiled by such a local phase change. The covariant derivative includes the gauge field as a correction term that compensates exactly for the change. So the replacement makes the free theory consistent with the local symmetry, and in doing so it fixes how matter and gauge field interact. The strength of the interaction is set by a single number, the charge.

The word "minimal" refers to the fact that other interaction terms are also compatible with gauge symmetry, for example terms that couple a particle directly to the field strength. Minimal coupling leaves all of them out. In the Standard Model all couplings of matter to the gauge fields are of the minimal form, and the extra terms are suppressed at low energies.

An example of a prediction from minimal coupling alone: applied to the Dirac equation it yields the magnetic moment of the electron with g-factor equal to two, before quantum corrections.

Concrete

Classical particle. A particle with charge $q$ in a scalar potential $\phi$ and vector potential $\vec A$ has the Lagrangian and Hamiltonian

$L = \frac{1}{2} m \dot{\vec x}^2 - q\phi + q\, \dot{\vec x}\cdot\vec A , \qquad H = \frac{(\vec p - q\vec A)^2}{2m} + q\phi .$

Here $\vec p = m\dot{\vec x} + q\vec A$ is the canonical momentum. The Hamiltonian is obtained from the free one by the replacement $\vec p \to \vec p - q\vec A$ , $H \to H - q\phi$ . The Euler-Lagrange equations give the Lorentz force $m\ddot{\vec x} = q(\vec E + \dot{\vec x}\times\vec B)$ with $\vec E = -\nabla\phi - \partial_t\vec A$ and $\vec B = \nabla\times\vec A$ .

Quantum mechanics. With $\vec p = -i\hbar\nabla$ and $E = i\hbar\partial_t$ the same replacement gives the Schrödinger equation in an electromagnetic field

$i\hbar\,\partial_t\psi = \left[ \frac{(-i\hbar\nabla - q\vec A)^2}{2m} + q\phi \right]\psi .$

It is invariant under the gauge transformation

$\vec A \to \vec A + \nabla\chi, \qquad \phi \to \phi - \partial_t\chi, \qquad \psi \to e^{iq\chi/\hbar}\,\psi ,$

because $(-i\hbar\nabla - q\vec A)$ acting on $e^{iq\chi/\hbar}\psi$ produces an extra $q\nabla\chi\, e^{iq\chi/\hbar}\psi$ that the shift of $\vec A$ cancels, and similarly for the time derivative.

Relativistic form. With metric signature $(+,-,-,-)$ , $A^\mu = (\phi/c, \vec A)$ and $p_\mu = i\hbar\partial_\mu$ , the replacement $p_\mu \to p_\mu - qA_\mu$ reads

$\partial_\mu \to D_\mu = \partial_\mu + \frac{iq}{\hbar} A_\mu ,$

and under $\psi \to e^{iq\chi/\hbar}\psi$ , $A_\mu \to A_\mu - \partial_\mu\chi$ we get $D_\mu\psi \to e^{iq\chi/\hbar} D_\mu\psi$ . Setting $\hbar = c = 1$ , the free Dirac Lagrangian $\bar\psi(i\gamma^\mu\partial_\mu - m)\psi$ becomes

$\mathcal L = \bar\psi(i\gamma^\mu D_\mu - m)\psi - \frac14 F_{\mu\nu}F^{\mu\nu} = \bar\psi(i\gamma^\mu\partial_\mu - m)\psi - q\,\bar\psi\gamma^\mu\psi\, A_\mu - \frac14 F_{\mu\nu}F^{\mu\nu} ,$

so the interaction is $-j^\mu A_\mu$ with the conserved current $j^\mu = q\bar\psi\gamma^\mu\psi$ . Squaring the Dirac operator with $D_\mu$ in place of $\partial_\mu$ and taking the non-relativistic limit gives the Pauli equation, with the spin term $-\frac{q\hbar}{2m}\vec\sigma\cdot\vec B$ , which is the statement $g = 2$ .

A non-minimal term. The Pauli term $\bar\psi\sigma^{\mu\nu}\psi F_{\mu\nu}$ , with $\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu,\gamma^\nu]$ , is gauge invariant as well and shifts the magnetic moment away from $g = 2$ . It has mass dimension 5, so its coefficient has dimension of inverse mass and the term is not renormalizable. This is the connection to renormalizability made in the quote under "Why is it interesting?".

For a non-Abelian group with generators $T^a$ the covariant derivative is $D_\mu = \partial_\mu + i g A_\mu^a T^a$ (sign conventions differ between textbooks), and minimal coupling works the same way.

Abstract

Let $P \to M$ be a principal $G$ -bundle with connection $A$ , and let matter fields be sections of an associated vector bundle $E = P\times_\rho V$ for a representation $\rho$ of $G$ . The connection induces a covariant derivative $\nabla^A$ on sections of $E$ , locally $\nabla^A = d + \rho_*(A)$ where $\rho_*$ is the induced Lie algebra representation (with a factor of $i$ and the coupling constant depending on conventions). Minimal coupling is the prescription: take a Lagrangian for $V$ -valued fields that is invariant under global $G$ transformations, and replace $d$ by $\nabla^A$ everywhere. The result is automatically invariant under gauge transformations, since $\nabla^A$ transforms covariantly under bundle automorphisms. The gauge field itself enters through its curvature $F = dA + A\wedge A$ in the Yang-Mills term.

The prescription is not unique in general. Covariant derivatives do not commute, $[\nabla_\mu,\nabla_\nu] = \rho_*(F_{\mu\nu})$ , so when the free Lagrangian has more than one derivative on a field, different orderings of the same expression differ by terms proportional to $F$ . More generally every gauge invariant term built with $F$ explicitly, such as the Pauli term, is consistent with the symmetry. Minimal coupling means setting all such terms to zero at the scale where the theory is defined. Loop corrections still produce the effects of such terms. In a renormalizable, minimally coupled theory such as QED the coefficients of the irrelevant ones come out finite and calculable; the anomalous magnetic moment of the electron is the standard example.

For a classical point particle the prescription has a geometric form. The charged particle action is $S = -m\int ds - q\int_\gamma A$ , the integral of the connection one-form along the worldline. On the Hamiltonian side, writing the canonical form in terms of the kinetic momentum $\pi_i = p_i - qA_i$ gives

$dp_i\wedge dx^i = d\pi_i\wedge dx^i + q\, dA ,$

so one can equivalently keep the free Hamiltonian $\pi^2/2m$ and add the magnetic two-form $q F$ (here $F = dA$ on space) to the symplectic structure. This version is manifestly gauge invariant and does not need a global potential, which matters for magnetic monopole backgrounds.

The same idea applied to gravity replaces $\eta_{\mu\nu}$ by $g_{\mu\nu}$ and $\partial_\mu$ by the Levi-Civita covariant derivative. It has the same kind of ambiguity, for a scalar for example the curvature coupling $\xi R\phi^2$ . For fields of spin $3/2$ and higher, minimal coupling to electromagnetism leads to inconsistencies such as acausal propagation (Velo and Zwanziger), so for these fields the minimal prescription is not sufficient.

Why is it interesting?

Why minimal coupling? This requirement arises indirectly, as a consequence of the general difficulty of constructing consistent interacting relativistic quantum field theories in four space-time dimensions. The perturbative construction of such theories involves regularization and renormaliza- tion. If we want to have a closed theory, in which the limiting effect of the regulator can be absorbed into a finite number of renormalization parameters, we must restrict ourselves to renormalizable couplings, that is interaction vertices whose coefficients have non-negative mass dimension (in units with ~ = c = 1, of course). Otherwise the insertion of these vertices will, in general, be accompanied by positive powers of the cutoff mass, and taking that mass to infinity will be counterproductive. This means that the vertices themselves must be constructed from fields whose total mass dimension does not exceed four. This criterion, when combined with gauge invariance, greatly restricts the possibilities, and in particular enforces minimal coupling for the gauge fields. Yang-Mills Theory In, Beyond, and Behind Observed Reality by Frank Wilczek

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Authors: Jakob Schwichtenberg